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Description

Recent analysis of truly huge quantities of empirical data suggests that classic economic theories not only fail for a few outliers, but that there occur similar outliers of every possible size. In fact, if one analyzes only a small data set (say 104 data points), then outliers appear to occur as “rare events.” However, when we analyze orders of magnitude more data (108 data points!), we find orders of magnitude more outliers - so ignoring them is not a responsible option, and studying their properties becomes a realistic goal. We find that the statistical properties of these “outliers” are identical to the statistical properties of everyday fluctuations. For example, a histogram giving the number of fluctuations of a given magnitude x for fluctuations ranging in magnitude from everyday fluctuations to extremely rare fluctuations that occur with a probability of only 10−8 is a perfect straight line in a double-log plot.
Two unifying principles that underlie much of the finance analysis we will present are scale invariance and universality [ R. N. Mantegna/HES, Introduction to Econophysics: Correlations & Complexity in Finance (Cambridge U. Press, 2000)]. Scale invariance is a property not about algebraic equations but rather about functional equations, which have as their solutions not numbers but rather functional forms - power laws.
The key idea of universality is that the identical set of “scaling laws” hold across diverse markets, and over diverse time periods.
We demonstrate the principles of scaling and universality by describing very recent unpublished work [HES/T. Preis/J. J. Schneider “New Laws Describing Trend Switching Processes in Financial Markets”, submitted]. For an intriguing variety of switching processes in nature, the underlying complex system abruptly changes at a specific “phase transition” point from one state to another in a highly discontinuous fashion. Examples of phase transitions range from magnetism in statistical physics to physiology and macroscopic social phenomena. Financial market fluctuations are characterized by many abrupt switchings on very short time scales from increasing “microtrends” to decreasing “microtrends”—and vice versa. We ask whether these ubiquitous switching processes have quantifiable features analogous to those present in phase transitions, and find striking scale-free behavior of the time intervals between transactions both before and after the switching occurs. We interpret our findings as being consistent with time-dependent collective behavior of financial market participants. We test the possible universality of our result by performing a parallel analysis of transaction volume fluctuations.

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